309.sagews

# Mathematical Methods /Numerical codes
#http://www1.uprh.edu/rbaretti
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysics.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart2.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart3.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart4.htm
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysicsPart5.htm




diff(x^3,x,2)

6*x

f=x^2
diff(f,x)

2*x

integral(f,x)

x^3/3

integral(f,x,0,3)

9

#x,y=var('x,y')
# plot3d(16-x^2-y^2,(x,-2,2),(y,-2,2))

#y=plot(sin(x),0,pi)    
#show(y)

y=plot(exp(-x),0,4)
show(y)

plot ((x^2+1),x,-2,2).show(xmin=-3,xmax=3,ymin=0.,ymax=6)

integral(exp(-2*x),x,0,20)

1/2 - e^-40/2

# two linear equations
x, y = var('x, y')
solve([8*x + y == 3, -x +7*y == 0],x,y)

[[x == 7/19, y == 1/19]]

# example of Kirchoff eqs fro three currents
i1,i2,i3=var('i1,i2,i3')
solve([8*i1 + i2 +i3== 3, -i1 +7*i2 +i3== 0,i1+i2+i3==1],i1,i2,i3)

[[i1 == 2/7, i2 == -1/14, i3 == 11/14]]

# To solve the equation x''+9*x=0: 

t = var('t')    # define a variable t
x = function('x',t)   # define x to be a function of that variable
DE = lambda y: diff(y,t,2) +9*y 
desolve(DE(x(t)), [x,t])

k1*sin(3*t) + k2*cos(3*t)

# To solve the equation x''+ 9*x=0: 

t = var('t')    # define the variable t
x=function('x',t)
DE=lambda x:diff(x,t,2)+diff(x,t)+ 9*x
desolve (DE(x(t)),[x,t])

e^(-(t/2))*(k1*sin(sqrt(35)*t/2) + k2*cos(sqrt(35)*t/2))

var('x,y,k ,L')
x=10
f=2*k*L/(x^2+y^2)^(1/2)
integral(f,y,0,100*x)

2*arcsinh(100)*k*L

# the R declaration defines the decimal result in terms of bits
R=RealField(30)
f=1/(5+x^2)^(1/2)
g=integral(f,x,0,5)
R(integral(f,x,0,5))

1.5444850

var('x, y')
y = integral(sqrt(1 + x^2), x, 0, 2)
print y
#  n(y,20)

arcsinh(2) + 2 sqrt(5) -------------------- 2

n((arcsinh(2) + 2*sqrt(5))/2,20)

2.9579

# 2x2 matrix 
 A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]); 
print(A)

Syntax Error: A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]);

plot (.434*log(x),x,1,10).show(xmin=0,xmax=10,ymin=0.,ymax=1)

a=pi;p=3;
y=plot((a/pi)^(1/4)*cos(p*x)*exp(-a*x^2/2),-3,3)
show(y)

#cos(px/h') (α/π )1/4 exp ( - αx2 /2)

var('x,n '); n=9;
f=n/(pi*(1+(n*x)^2));
integral(f,x,-10,10)

2*arctan(90)/pi

R=RealField(30)
R(2*arctan(90)/pi)

0.99292674

var('x,mu,sigma');sigma=.001;mu=0;eta=20;
gauss=(1/sigma)*(1/(2*pi)^(1/2))*exp(-(x-mu)^2/(2*sigma^2));

y=plot(-eta*gauss,x,-3*sigma,3*sigma)
show(y)

var('x,mu,sigma');sigma=.001;mu=0;eta=20;
gauss=(1/sigma)*(1/(2*pi)^(1/2))*exp(-(x-mu)^2/(2*sigma^2));
integral(gauss,x,-oo,oo)

0.707106781187*sqrt(2)

var('t');
y=plot(10*t*exp(-t^2),0,2.5)
show(y)

#f=2*x^2+1 ;
#integral(f,x,2,5)
K=81, m=0.6 , 
v=(2*K/m)^(1/2)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/11/code/6.py", line 8, in <module> K=_sage_const_81 , m=_sage_const_0p6 , ValueError: need more than 1 value to unpack

K=81; m=0.6 ; 
n(2*K/m)^(1/2)

16.4316767251550

var('t')
x=5*t^2+3*t;
v=diff(x,t);

10*t + 3

var('x');
y=5*x^2;v=diff(y,x)
x=2;
show(v)

10 \, x

r = 4
area = pi*r^2
area,area.n()

(16*pi, 50.2654824574367)

var('x');
y=5*x^2;
v=diff(y,x)

v

10*x

v=10*x
x=3
v,v.n()

(30, 30.0000000000000)

var('r');
f=27- 18*r + 2*r^2 ;
y=plot(f,r,0,10)
show (y)

f=r^3*exp(-r);
exp(r)*diff(f,r,3)

-(r^3*e^(-r) - 9*r^2*e^(-r) + 18*r*e^(-r) - 6*e^(-r))*e^r

#( 27- 18 r + 2 r2 )2 exp( -2r/3) 4π r2 
f=(27- 18*r + 2*r^2)^2*4*pi*r^2*exp(-2*r/3);
integral(f,r,0,oo)

19683*pi

81^2*3

19683

#{ 81(3 π) }-1/2  ( 27- 18 r + 2 r2 )2 exp( -2r/3)
f=(81*3*pi)^(-1/2)*( 27- 18*r + 2*r^2)*exp( -r/3)
plot(f,r,0,20)

u=var('u');
#(1/2)*( 5*x^3 - 3*x) 

#f=(1/2)*(5*(2*u-1)^3 -3*(2*u-1) );
f1=5*(2*u-1)^3;
f1

5*(2*u - 1)^3

u=var('u');
p2(u)= 6*u^2 -6*u + 1; p3(u)=20*u^3 -30*u^2 +12*u -1;
integral(p2(u)*p3(u),u,0,1)

0

var('a');Z=2;En=(5/4)*(a^2/2 -Z*a); E1= 2*(a^2/2 -Z*a);E2= 2*(1/4)*(a^2/2 -Z*a); 
e=(5/4)*(a^2 /2  -Z*a)  + (17/81)*a + (8.936E-2* a)^2 /(En -E1)+ (8.582e-3*a)^2/(En -E2);
plot(e,a,1.4,2.2)

integral(1/(1+x^2),x)

arctan(x)

f(x)=exp(-x^2);
integral(f(x),x,-oo,oo)

sqrt(pi)

n( pi^(1/2) )

1.77245385090552